How mathematicians think using ambiguity, contradiction, and paradox to create mathematics
To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive...
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| Hlavní autor: | |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Princeton, N.J. ; Woodstock
Princeton University Press
2010
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| Vydání: | 1 |
| Témata: | |
| ISBN: | 0691127387, 9780691145990, 9780691127385, 0691145997, 1400833957, 9781400833955 |
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| Abstract | To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics,How Mathematicians Thinkreveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results.
Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure.
The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, andHow Mathematicians Thinkprovides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory?
Ultimately,How Mathematicians Thinkshows that the nature of mathematical thinking can teach us a great deal about the human condition itself. |
|---|---|
| AbstractList | To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself. No detailed description available for "How Mathematicians Think". To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics,How Mathematicians Thinkreveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, andHow Mathematicians Thinkprovides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately,How Mathematicians Thinkshows that the nature of mathematical thinking can teach us a great deal about the human condition itself. To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even algorithmically--from one black-and-white deduction to another. Yet mathematicians often describe their most important breakthroughs as creative, intuitive responses to ambiguity, contradiction, and paradox. A unique examination of this less-familiar aspect of mathematics, How Mathematicians Think reveals that mathematics is a profoundly creative activity and not just a body of formalized rules and results. Nonlogical qualities, William Byers shows, play an essential role in mathematics. Ambiguities, contradictions, and paradoxes can arise when ideas developed in different contexts come into contact. Uncertainties and conflicts do not impede but rather spur the development of mathematics. Creativity often means bringing apparently incompatible perspectives together as complementary aspects of a new, more subtle theory. The secret of mathematics is not to be found only in its logical structure. The creative dimensions of mathematical work have great implications for our notions of mathematical and scientific truth, and How Mathematicians Think provides a novel approach to many fundamental questions. Is mathematics objectively true? Is it discovered or invented? And is there such a thing as a "final" scientific theory? Ultimately, How Mathematicians Think shows that the nature of mathematical thinking can teach us a great deal about the human condition itself. |
| Author | Byers, William |
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| Copyright | 2007 Princeton University Press |
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| Keywords | Scientific theory Subset Series (mathematics) Rational number Counterexample Logical reasoning Result Complexity Gödel's incompleteness theorems Mathematical theory Mathematical induction Calculation Axiom Contradiction Paradox Addition Real number Approximation Cardinality Consciousness Rationality Conjecture Scientist Inference Continuous function Principle Algorithm Thought Existential quantification Cardinal number Irrational number Informal mathematics Derivative Philosophy of science Summation Ambiguity Natural number Sign (mathematics) Philosophy of mathematics Line segment The Unreasonable Effectiveness of Mathematics in the Natural Sciences Theory Science Mathematics Notation Reason Actual infinity Mathematical proof Pure mathematics Emergence Theorem Analogy Logic Proof by contradiction Euclidean geometry Equation Quantity Dimension Infinitesimal Mathematician Complex number Variable (mathematics) Geometry Integer David Hilbert Parity (mathematics) Quantum mechanics Randomness Mathematical practice Parallel postulate |
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| Notes | Originally published: 2007 Includes bibliographical references (p. 399-405) and index "Sixth printing, and first paperback printing, 2010" -- T. p. verso |
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| Snippet | To many outsiders, mathematicians appear to think like computers, grimly grinding away with a strict formal logic and moving methodically--even... No detailed description available for "How Mathematicians Think". |
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| SubjectTerms | Actual infinity Addition Algorithm Ambiguity Analogy Approximation Axiom Calculation Cardinal number Cardinality Complex number Complexity Conjecture Consciousness Continuous function Contradiction Counterexample David Hilbert Derivative Dimension Emergence Equation Euclidean geometry Existential quantification Geometry Gödel's incompleteness theorems History & Philosophy Inference Infinitesimal Informal mathematics Integer Irrational number Line segment Logic Logical reasoning Mathematical induction Mathematical practice Mathematical proof Mathematical theory Mathematician Mathematicians Mathematicians -- Psychology MATHEMATICS Mathematics -- Philosophy Mathematics -- Psychological aspects MATHEMATICS / History & Philosophy Natural number Notation Paradox Parallel postulate Parity (mathematics) PDZ Philosophy Philosophy of mathematics Philosophy of science Principle Proof by contradiction Psychological aspects Psychology Pure mathematics Quantity Quantum mechanics Randomness Rational number Rationality Real number Reason Result Science Scientific theory Scientist Series (mathematics) Sign (mathematics) Subset Summation The Unreasonable Effectiveness of Mathematics in the Natural Sciences Theorem Theory Thought Variable (mathematics) |
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| Subtitle | using ambiguity, contradiction, and paradox to create mathematics |
| TableOfContents | How mathematicians think: using ambiguity, contradiction, and paradox to create mathematics -- Contents -- Acknowledgments -- Introduction: Turning on the Light -- Section I: The Light of Ambiguity -- Chapter 1: Ambiguity in Mathematics -- Chapter 2: The Contradictory in Mathematics -- Chapter 3: Paradoxes and Mathematics: Infinity and the Real Numbers -- Chapter 4: More Paradoxes of Infinity: Geometry, Cardinality, and Beyond -- Section II: The Light as Idea -- Chapter 5: The Idea as an Organizing Principle -- Chapter 6: Ideas, Logic, and Paradox -- Chapter 7: Great Ideas -- Section III: The Light and the Eye of the Beholder -- Chapter 8: The Truth of Mathematics -- Chapter 9: Conclusion: Is Mathematics Algorithmic or Creative? -- Notes -- Bibliography -- Index. Front Matter Table of Contents Acknowledgments INTRODUCTION CHAPTER 1: Ambiguity in Mathematics CHAPTER 2: The Contradictory in Mathematics CHAPTER 3: Paradoxes and Mathematics: CHAPTER 4: More Paradoxes of Infinity: CHAPTER 5: The Idea as an Organizing Principle CHAPTER 6: Ideas, Logic, and Paradox CHAPTER 7: Great Ideas CHAPTER 8: The Truth of Mathematics CHAPTER 9: Conclusion: Notes Bibliography Index Cover -- Title -- Copyright -- Contents -- Acknowledgments -- INTRODUCTION: Turning on the Light -- SECTION I: THE LIGHT OF AMBIGUITY -- CHAPTER 1 Ambiguity in Mathematics -- CHAPTER 2 The Contradictory in Mathematics -- CHAPTER 3 Paradoxes and Mathematics: Infinity and the Real Numbers -- CHAPTER 4 More Paradoxes of Infinity: Geometry, Cardinality, and Beyond -- SECTION II: THE LIGHT AS IDEA -- CHAPTER 5 The Idea as an Organizing Principle -- CHAPTER 6 Ideas, Logic, and Paradox -- CHAPTER 7 Great Ideas -- SECTION III: THE LIGHT AND THE EYE OF THE BEHOLDER -- CHAPTER 8 The Truth of Mathematics -- CHAPTER 9 Conclusion: Is Mathematics Algorithmic or Creative? -- Notes -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z Section I. The Light of Ambiguity -- INTRODUCTION. Turning on the Light Chapter 5. The Idea as an Organizing Principle Chapter 3. Paradoxes and Mathematics: Infinity and the Real Numbers Chapter 1. Ambiguity in Mathematics Chapter 6. Ideas, Logic, and Paradox Index - Chapter 2. The Contradictory in Mathematics Chapter 7. Great Ideas Chapter 8. The Truth of Mathematics / Section III. The Light and the Eye of the Beholder -- Chapter 9. Conclusion: Is Mathematics Algorithmic or Creative? Section II. The Light as Idea -- Contents Chapter 4. More Paradoxes of Infinity: Geometry, Cardinality, and Beyond Acknowledgments Introduction Frontmatter -- Notes Bibliography |
| Title | How mathematicians think |
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